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The KNTZ trick from arborescent calculus and the structure of the differential expansion

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Abstract

The recently proposed Kameyama–Nawata–Tao–Zhang (KNTZ ) trick completed the long search for exclusive Racah matrices \( \kern2.3pt\overline{\vphantom{S}\kern3.70831pt}\kern-6.00832pt{S} \) and \(S\) for all rectangular representations. The success of this description is a remarkable achievement of modern knot theory and classical representation theory, which was initially considered a tool for knot calculus but instead turned out to be its direct beneficiary. We show that this approach in fact consists in converting the arborescent evolution matrix \( \kern2.3pt\overline{\vphantom{S}\kern3.70831pt}\kern-6.00832pt{S} \kern1.5pt\overline{\vphantom{{T}}\kern5.23265pt}\kern-6.73265pt{T} ^2 \kern2.3pt\overline{\vphantom{S}\kern3.70831pt}\kern-6.00832pt{S} \) into the triangular form \( \mathcal{B} \), and we demonstrate how this works and show how the previous puzzles and miracles of the differential expansions look from this standpoint. Our conjecture for the form of the triangular matrix \( \mathcal{B} \) in the case of the nonrectangular representation \([3,1]\) is completely new. No calculations are simplified in this case, but we explain how it all works and what remains to be done to completely prove the conjecture. The discussion can also be useful for extending the method to nonrectangular cases and for the related search for gauge-invariant arborescent vertices. As one more application, we present a puzzling, but experimentally supported, conjecture that the form of the differential expansion for all knots is completely described by a particular case of twist knots.

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Funding

This research was supported in part by a grant from the Russian Science Foundation (Project No. 16-12-10344).

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  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. Yu. Morozov

  2. Institute for Information Transmission Problems, Moscow, Russia

    A. Yu. Morozov

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia

    A. Yu. Morozov

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Appendix

In this appendix, we present the fragment of \( \mathcal{B} ^{ \mathrm{rel} }\) that includes all the representations in \([3,1]\times \overline {[3,1]}\). We described its role and applications in Sec. 10.2. The full \(10{\times}10\) matrix \( \mathcal{B} \) in these coordinates also has a block form: the blocks of \( \mathcal{B} ^{ \mathrm{rel} }_{[3,1]}\) are

$$\begin{aligned} \, &\left(\begin{array}{c|c|c|c|c|c|c|c|c|c|c} &\kern-8.05pt& \varnothing &\kern-8.05pt&[1]_z&[2]_z&\kern-8.05pt&[1]_y&[2]_y&[1]_x&[2]_x\\\hline &\kern-8.05pt&&\kern-8.05pt&&&\kern-8.05pt&&&&\\[-4.25mm]\hline \varnothing &\kern-8.05pt&1&\kern-8.05pt&&&\kern-8.05pt&&&&\\ {}[1]_z&\kern-8.05pt&0&\kern-8.05pt&A^2&&\kern-8.05pt&&&&\\\hline &\kern-8.05pt&&\kern-8.05pt&&&\kern-8.05pt&&&&\\[-4.25mm]\hline {}[2]_z&\kern-8.05pt&0&\kern-8.05pt&-q^2A^3&q^4A^4&\kern-8.05pt&&&&\\ {}[1]_y&\kern-8.05pt&-A^2&\kern-8.05pt&0&0&\kern-8.05pt&A^2&&&\\ {}[2]_y&\kern-8.05pt&q^2A^4&\kern-8.05pt& \underline {-q^3A^4[N-2]}&0&\kern-8.05pt&-[2]q^3A^4&q^4A^4&&\\ {}[1]_x&\kern-8.05pt&-A^2&\kern-8.05pt&0&0&\kern-8.05pt&0&0&A^2&\\ {}[2]_x&\kern-8.05pt&q^2A^4&\kern-8.05pt& \underline {-q^3A^4[N-2]}&0&\kern-8.05pt & \underline {-\frac{q^3A^4}{[2]}}&0& \underline {-\frac{q^3[3]A^4}{[2]}}&q^4A^4\\\hline {}[3]&\kern-8.05pt&-q^6A^6&\kern-8.05pt&q^8A^6[2][N-2]&q^{11}A^7[N-2]&\kern-8.05pt &q^8A^6&-\frac{q^{10}A^6}{[3]}&\frac{[4]q^8A^6}{[2]}&-\frac{[4][2]q^{10}A^6}{[3]} \\\hline {}[1,1]&\kern-8.05pt&\frac{A^4}{q^2}&\kern-8.05pt&\frac{[2]A^4[N+3]}{q^3}&0&\kern-8.05pt &-\frac{[2]A^4}{q^3[3]}&0&-\frac{[4]A^4}{q^3[3]}&0 \\\hline {}[2,1]&\kern-8.05pt&-A^6&\kern-8.05pt&{-[3]A^5}-\frac{A^6[N+4]}{q}&-\frac{q^2A^7[N+4]}{[2]} &\kern-8.05pt&A^6&-\frac{qA^6}{[3][2]}&\frac{[4]A^6}{[2]}&-\frac{[4]qA^6}{[3]} \\\hline {}[3,1]&\kern-8.05pt&q^4A^8&\kern-8.05pt&q^4[4][2]A^7&q^{7}[4]A^8&\kern-8.05pt&-\frac{[4]q^5A^8}{[3]} &\frac{[4]q^7A^8}{[3]^2}&-\frac{[4]^2q^5A^8}{[3][2]}&\frac{[4]^2[2]q^7A^8}{[3]^2} \end{array}\right), \\[4mm] &\begin{aligned} \, &\left(\begin{array}{c|c|c|c|c|c} &\kern-8.05pt& \varnothing &\kern-8.05pt&[1]_z&[2]_z\\\hline &\kern-8.05pt&&\kern-8.05pt&&\\[-4.25mm]\hline X_2&\kern-8.05pt&-A^6&\kern-8.05pt&-[3]A^5-\frac{A^6[N+4]}{q}+\frac{A^6[N]}{\{q\}} &-\frac{q^2A^6[N+4][N+1]}{[2]} \\ X_3&\kern-8.05pt&q^4A^8&\kern-8.05pt&q^4[4][2]A^7-\frac{q^5A^8[N+1]}{\{q\}} &q^7[4]A^8-\frac{q^6A^8[N+2][N+1]}{[2]}-\frac{q^3A^7[N+1]}{\{q\}} \end{array}\right), \\[4mm] &\left(\begin{array}{c|c|c|c} &\kern-8.05pt&[1]_y&[2]_y\\\hline &\kern-8.05pt&&\\[-4.25mm]\hline X_2&\kern-8.05pt&A^6-\frac{A^5[N]}{q[3][2]\{q\}}&-\frac{qA^5[N+1]}{[3][2]} \\ X_3&\kern-8.05pt&-\frac{[4]q^5A^8}{[3]}+\frac{q^4A^7[N+1]}{[3][2]\{q\}} &\frac{[4]q^7A^8}{[3]^2}-\frac{[4]q^4A^7[N+1]}{[3]^2[2]^2\{q\}} \end{array}\right), \\[4mm] &\left(\begin{array}{c|c|c|c|c|c} &\kern-8.05pt&X_2&X_3&\kern-8.05pt&\dots\\\hline &\kern-8.05pt&&&\kern-8.05pt&\\[-4.25mm]\hline X_2&\kern-8.05pt&\frac{[4]A^6}{[2]}+\frac{A^5[N]}{q[3][2]\{q\}}&-\frac{qA^5[N+4]}{[3]} &\kern-8.05pt&\\ X_3&\kern-8.05pt&-\frac{[4]^2q^5A^8}{[3][2]}-\frac{q^4A^7[N+1]}{[3][2]]\{q\}} &\frac{[4]^2[2]q^7A^8}{[3]^2}+\frac{q^3A^7[N+1]}{[2]}+ \frac{[4]q^4A^7[N+1]}{[3]^2[2]^2\{q\}} &\kern-8.05pt&\\\hline &\kern-8.05pt&&&\kern-8.05pt&\\[-4.25mm]\hline \vdots&\kern-8.05pt&&&\kern-8.05pt&\end{array}\right), \end{aligned} \\[4mm] &\left(\begin{array}{c|c|c|c|c|c|c|c|c|c|c} &\kern-8.05pt&[3]&\kern-8.05pt&[1,1]&[2,1]&\kern-8.05pt&[3,1]&X_2&X_3&\dots\\\hline &\kern-8.05pt&&\kern-8.05pt&&&\kern-8.05pt&&&&\\[-4.25mm]\hline {}[3]&\kern-8.05pt&q^{12}A^6&\kern-8.05pt&&&\kern-8.05pt&&& \\\hline {}[1,1]&\kern-8.05pt&0&\kern-8.05pt&\frac{A^4}{q^4}&&\kern-8.05pt&&& \\ {}[2,1]&\kern-8.05pt&0&\kern-8.05pt&-\frac{[3]A^6}{q[2]}&A^6&\kern-8.05pt&&&& \\ {}[3,1]&\kern-8.05pt&-\frac{[4]q^9A^8}{[3]}&\kern-8.05pt&\frac{[4]q^4A^8}{[2]}&-\frac{[4][2]q^6A^8}{[3]}&\kern-8.05pt&q^8A^8&&& \\\hline X_2&\kern-8.05pt&0&\kern-8.05pt&-\frac{A^5[N-3]}{q[2]}&0&\kern-8.05pt&0&A^4&& \\ X_3&\kern-8.05pt&-\frac{q^8A^7[N+5]}{[3]}&\kern-8.05pt&-\frac{q^3A^7[N-3]}{[2]}&\frac{q^5A^7[N-1][N-3]}{[3][N]}&\kern-8.05pt&0&-\frac{q^5A^6[N+1]}{[N]}&q^6A^6 \\\hline &\kern-8.05pt&&\kern-8.05pt&&&\kern-8.05pt&&&&\\[-4.25mm]\hline \vdots&\kern-8.05pt&&\kern-8.05pt&&&\kern-8.05pt&&&&\end{array}\right). \end{aligned}$$

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Morozov, A.Y. The KNTZ trick from arborescent calculus and the structure of the differential expansion. Theor Math Phys 204, 993–1019 (2020). https://doi.org/10.1134/S0040577920080036

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